Symmetric Dyck Paths and Hooley’s \(\varDelta \)-Function

  • José Manuel Rodríguez CaballeroEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10432)


Hooley [6] introduced the function
$$ \varDelta (n) := \max _{u \in \mathbb {R}}\# \left\{ d | n: \quad u < \log d \leqslant u+1 \right\} , $$
where \(\log \) is the natural logarithm. Changing the base of the logarithm from e to an arbitrary real number \(\lambda > 1\), we define
$$ \varDelta _{\lambda }(n) := \max _{u \in \mathbb {R}}\# \left\{ d | n:\quad u < \log _{\lambda } d \leqslant u+1 \right\} . $$
The aim of this paper is to express \(\varDelta _{\lambda }(n)\) as the height of a symmetric Dyck path defined in terms of the distribution of the divisors of n.


Dyck path Palindrome Hooley’s \(\varDelta \)-function 



The author thanks S. Brlek, C. Kassel and C. Reutenauer for they valuable comments and suggestions concerning this research. Also, the author want to express his gratitude to H. F. W. Höft for the useful exchanges of information.


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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Université du Québec à MontréalMontréalCanada

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